A Rank Theorem of Operators between Banach Spaces

Abstract

Suppose that E and F are two Banach spaces and that B(E, F) is the space of all bounded linear operators from E to F. Let T₀∈B(E, F) with a generalized inverse T₀ ⁺∈B(F, E). This paper shows that, for every T∈B(E, F) with ‖T₀ ⁺ (T−T₀)‖<1, B ≡ (I + T₀ ⁺(T−T₀))⁻¹T₀ ⁺ is a generalized inverse of T if and only if (I−T₀ ⁺T₀)N(T) = N(T₀), where N(·) stands for the null space of the operator inside the parenthesis. This result improves a useful theorem of Nashed and Cheng and further shows that a lemma given by Nashed and Cheng is valid in the case where T₀ is a semi-Fredholm operator but not in general.

Keywords

rank theorem / generalized inverse / linear semi-Fredholm operator / Banach space / primary 47A05 / secondary 47A53

References

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